Calculus II PLTLFall 2014Worksheet 10These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Pairs) (a) Explain, without formulas, what a power series is.(b) Explain, witho ut formulas, what the interval of convergence of a power seri es is.(c) List the six possible forms of the interval of convergence.(d) Explain, without formulas, what the phrase “radius of convergence of a powerseries ” means.(e) How are power series different from series you have been working with in previousweeks?2) (Scribe) Find the interval and r adius of convergence of the power seriesP∞n=0(2x−1)nn2+1.Be sure to check each endpoint o f the interval for convergence.3) (Round Robin) Let P5(x) = c0+ c1x + c2x2+ c3x3+ c4x4+ c5x5be the Taylorpolynomial of degree 5 for the function f(x) = x2e−2xcentered about the origin.(a) How do you calculate the coeffi cients c0, c1, etc.?(b) What is P5(1)?(c) What is the value of f(1)?(d) The graph below shows both P5(x) and f(x). Would you feel comfortable using itto approximate the value of f(0.05), for example? What is the advantage of using P5(x)instead of f(x)?4) (Pairs) (a) Show that the seriesP∞n=1anwith an= 1/ n2converges u sing the IntegralTest.(b) Suppose that s =P∞n=1an. Use the remainder estimate for the integral test tofind the smallest integer n such that Rn= |s − sn| ≤ 0.01, where sn=Pnk=1an.5) (Round Robin) For parts A-D, determine which of the following is true:(a) Series I is convergent, but the others are divergent.(b) Series II is conver gent, but the others are divergent.(c) Series III is convergent, but the others are divergent.(d) Series I and Seri es II are convergent, but Series III is divergent.(e) Series I and Series III are convergent, but Series II is divergent.(f) Series II and Series III are convergent, but Series I is divergent.(g) All three s eries are convergent.(h) All three series are divergent.(i) There is not enough information to determine the convergence/divergence of eachof the series.A : (I )P∞n=1(−1)n−15n(II )P∞n=1(n+1)(n2−1)4n3−2n+1(III )P∞n=15(−4)n+232n+1B : (I )P∞n=21n(ln n)2(II )P∞n=12n+7n5n(III )P∞n=11√n3/2+n+1C : (I )P∞n=11n2+1(II )P∞n=1ln nn2+1(III )P∞n=11n2+3nHint : Compare ln n to√nD : (I )P∞n=1ennn(II )P∞n=1sin(1/n) (III )P∞n=113n−n−1Hint : limx→0sin(x)/x =
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